Homomorphisms between mapping class groups

Aramayona, Javier; Souto, Juan

doi:10.2140/gt.2012.16.2285

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Homomorphisms between mapping class groups

Javier Aramayona and Juan Souto

Geometry & Topology 16 (2012) 2285–2341

DOI: 10.2140/gt.2012.16.2285

Abstract

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g \geq 6$ and $Y$ has genus at most $2 g - 1$ ; in addition, suppose that $Y$ is not closed if it has genus $2 g - 1$ . Our main result asserts that every nontrivial homomorphism $Map (X) \to Map (Y)$ is induced by an embedding, ie a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every nontrivial endomorphism $Map (X) \to Map (X)$ is in fact an isomorphism.

Keywords

mapping class groups, superrigidity

Mathematical Subject Classification 2010

Primary: 20F34

Secondary: 57M07, 20F65

References

Publication

Received: 1 August 2011

Revised: 6 May 2012

Accepted: 27 July 2012

Published: 16 January 2013

Proposed: Cameron Gordon

Seconded: Colin Rourke, Walter Neumann

Authors

Javier Aramayona
School of Mathematics, Statistics and Applied Mathematics National University of Ireland Galway University Road Galway Ireland
http://www.maths.nuigalway.ie/~javier/
Juan Souto
Mathematics Department University of British Columbia 1984 Mathematics Road Vancouver, BC Canada V6T 1Z2
http://www.math.ubc.ca/~jsouto

Volume 16, issue 4 (2012)